What percentage of total flight time is spent at the top 10% of their jumping height?

Question

Basketball players, volleyball players, and other jumping athletes often seem to hang in the air for a long time. This is because they spend a comparatively high percentage of their total flight time in the highest 10% of their height.

Answer 1

The answer turns out to be 31.6 % of the total time of flight.

More precisely, 100 x (.1)^1/2

This can be worked out using the standard CAKE equations. The algebra is a bit messy.

It is easy to work out that the maximum height is ymax = v₀² /( 2g) and that the time to reach the maximum height is tau = v₀/g.

A key step is to introduce the variable T = t - tau . Then

y/ymax = 1 - ( g/v₀)² T²

Then letting R = the ratio T/tau

y/ymax = 1 - R²

Setting y/ymax = .9 gives R = sqrt( 1 - .9) = sqrt(.1)

Then multiply by 100 to get a percent.